Depending on the facts, a case may be dismissed if the police fail to read your Miranda rights. Under the law, law enforcement officers are required to give you these warnings before an in-custody interrogating. If they fail to do so, any statement that you make could be excluded from evidence. Typically, your attorney would file a motion to suppress any statement that was made during this interrogation, as well as any evidence obtained as a result of these statements.
In some cases, if the motion to suppress is granted, the prosecution will not have enough evidence to proceed with the charges against you. For example, consider a situation where the police interrogated you based on a hunch, and you confessed to the crime of murder — but there is absolutely no other evidence tying you to it. If your confession is suppressed (which means that it cannot be admitted into evidence) because the police did not read you your rights, then the case will likely be dismissed because there is no other evidence.
If there is other admissible evidence, the case will not likely be dismissed. In the same example as above, assume that the police also found the murder weapon in your home, and it has your fingerprints on it. There are also two witnesses to the killing. All of this evidence was found before the police interrogated you. While the prosecutor may have liked to have your confession as a piece of evidence, the charge will probably not be dismissed if the confession is suppressed because there is so much other evidence against you.